Adaptive Tracking and Parameter Estimation with Unknown High-Frequency Control Gains: A Case Study in Strictification

Transcription

1 Adaptive Tracking and Parameter Estimation with Unknown High-Frequency Control Gains: A Case Study in Strictification Michael Malisoff, Louisiana State University Joint with Frédéric Mazenc and Marcio de Queiroz Sponsored by AFOSR, NSF/DMS, and NSF/ECCS SIAM Conference on Control and Its Applications Baltimore, MD July 25, 2011

2 Adaptive Tracking and Estimation Problem

3 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters.

4 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters. ξ R = J (t, ξ R, Γ, u R ) t 0.

5 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters. ξ R = J (t, ξ R, Γ, u R ) t 0. Problem:

6 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters. ξ R = J (t, ξ R, Γ, u R ) t 0. Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΓ), ˆΓ = τ(t, ξ, ˆΓ) (2) that makes the Y = ( Γ, ξ) = (Γ ˆΓ, ξ ξ R ) system UGAS.

7 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters. ξ R = J (t, ξ R, Γ, u R ) t 0. Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΓ), ˆΓ = τ(t, ξ, ˆΓ) (2) that makes the Y = ( Γ, ξ) = (Γ ˆΓ, ξ ξ R ) system UGAS. Flight control, electrical and mechanical engineering, etc.

8 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters. ξ R = J (t, ξ R, Γ, u R ) t 0. Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΓ), ˆΓ = τ(t, ξ, ˆΓ) (2) that makes the Y = ( Γ, ξ) = (Γ ˆΓ, ξ ξ R ) system UGAS. Flight control, electrical and mechanical engineering, etc. Persistent excitation.

9 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters. ξ R = J (t, ξ R, Γ, u R ) t 0. Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΓ), ˆΓ = τ(t, ξ, ˆΓ) (2) that makes the Y = ( Γ, ξ) = (Γ ˆΓ, ξ ξ R ) system UGAS. Flight control, electrical and mechanical engineering, etc. Persistent excitation. Annaswamy, Narendra, Teel..

10 Adaptive Tracking and Estimation Problem Consider a suitably regular nonlinear system ξ = J (t, ξ, Γ, u) (1) with a smooth reference trajectory ξ R and a vector Γ of unknown constant parameters. ξ R = J (t, ξ R, Γ, u R ) t 0. Problem: Design a dynamic feedback with estimator u = u(t, ξ, ˆΓ), ˆΓ = τ(t, ξ, ˆΓ) (2) that makes the Y = ( Γ, ξ) = (Γ ˆΓ, ξ ξ R ) system UGAS. Flight control, electrical and mechanical engineering, etc. Persistent excitation. Annaswamy, Narendra, Teel.. Used nonstrict Lyapunov functions (LFs), Barbalat, LaSalle..

11 UGAS and Two of Sontag s Important Extensions

12 UGAS and Two of Sontag s Important Extensions Ẏ = G(t, Y ), Y X. (Σ)

13 UGAS and Two of Sontag s Important Extensions Ẏ = G(t, Y ), Y X. (Σ) Y (t) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) (UGAS)

14 UGAS and Two of Sontag s Important Extensions Ẏ = G(t, Y ), Y X. (Σ) Y (t) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) (UGAS) Show UGAS using nonstrict LFs, LaSalle,..

15 UGAS and Two of Sontag s Important Extensions Ẏ = G(t, Y ), Y X. (Σ) Y (t) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) (UGAS) Show UGAS using nonstrict LFs, LaSalle,.. Ẏ = G(t, Y, δ(t)), Y X. (Σ pert )

16 UGAS and Two of Sontag s Important Extensions Ẏ = G(t, Y ), Y X. (Σ) ( Y (t) γ 1 e t 0 t γ 2 ( Y (t 0 ) ) ) (UGAS) Show UGAS using nonstrict LFs, LaSalle,.. Ẏ = G(t, Y, δ(t)), Y X. (Σ pert ) ( Y (t) γ 1 e t 0 t γ 2 ( Y (t 0 ) ) ) + γ 3 ( δ [t0,t]) (ISS)

17 UGAS and Two of Sontag s Important Extensions Ẏ = G(t, Y ), Y X. (Σ) ( Y (t) γ 1 e t 0 t γ 2 ( Y (t 0 ) ) ) (UGAS) Show UGAS using nonstrict LFs, LaSalle,.. Ẏ = G(t, Y, δ(t)), Y X. (Σ pert ) ( Y (t) γ 1 e t 0 t γ 2 ( Y (t 0 ) ) ) + γ 3 ( δ [t0,t]) (ISS) γ 0 ( Y (t) ) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + t t 0 γ 3 ( δ(r) )dr (iiss)

18 UGAS and Two of Sontag s Important Extensions Ẏ = G(t, Y ), Y X. (Σ) ( Y (t) γ 1 e t 0 t γ 2 ( Y (t 0 ) ) ) (UGAS) Show UGAS using nonstrict LFs, LaSalle,.. Ẏ = G(t, Y, δ(t)), Y X. (Σ pert ) ( Y (t) γ 1 e t 0 t γ 2 ( Y (t 0 ) ) ) + γ 3 ( δ [t0,t]) (ISS) γ 0 ( Y (t) ) γ 1 ( e t 0 t γ 2 ( Y (t 0 ) ) ) + t t 0 γ 3 ( δ(r) )dr (iiss) Find γ i s by building certain strict LFs for Ẏ = G(t, Y, 0).

19 Our Work (Nonlinear Analysis TMA, 11)

20 Our Work (Nonlinear Analysis TMA, 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) (3) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s.

21 Our Work (Nonlinear Analysis TMA, 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) (3) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. Γ = (θ, ψ) = (θ 1,..., θ s, ψ 1,..., ψ s ) R p p s+s.

22 Our Work (Nonlinear Analysis TMA, 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) (3) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. Γ = (θ, ψ) = (θ 1,..., θ s, ψ 1,..., ψ s ) R p p s+s. The C 2 T -periodic reference trajectory ξ R = (x R, z R ) to be tracked is assumed to satisfy ẋ R (t) = f (ξ R (t)) t 0.

23 Our Work (Nonlinear Analysis TMA, 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) (3) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. Γ = (θ, ψ) = (θ 1,..., θ s, ψ 1,..., ψ s ) R p p s+s. The C 2 T -periodic reference trajectory ξ R = (x R, z R ) to be tracked is assumed to satisfy ẋ R (t) = f (ξ R (t)) t 0. Main PE Assumption:

24 Our Work (Nonlinear Analysis TMA, 11) We solved the adaptive tracking and estimation problem for { ẋ = f (ξ) (3) ż i = g i (ξ) + k i (ξ) θ i + ψ i u i, i = 1, 2,..., s. Γ = (θ, ψ) = (θ 1,..., θ s, ψ 1,..., ψ s ) R p p s+s. The C 2 T -periodic reference trajectory ξ R = (x R, z R ) to be tracked is assumed to satisfy ẋ R (t) = f (ξ R (t)) t 0. Main PE Assumption: positive definiteness of the matrices P i def = T 0 λ i (t)λ i (t) dt R (p i +1) (p i +1), 1 i s (4) where λ i (t) = (k i (ξ R (t)), ż R,i (t) g i (ξ R (t))) for each i.

25 Two Other Key Assumptions

26 Two Other Key Assumptions We know v f and a global strict LF V for { Ẋ = f ( (X, Z ) + ξr (t) ) f (ξ R (t)) Ż = v f (t, X, Z ) such that V and V have positive definite quadratic lower bounds near 0, (5)

27 Two Other Key Assumptions We know v f and a global strict LF V for { Ẋ = f ( (X, Z ) + ξr (t) ) f (ξ R (t)) Ż = v f (t, X, Z ) such that V and V have positive definite quadratic lower bounds near 0, and V and v f are T -periodic. (5)

28 Two Other Key Assumptions We know v f and a global strict LF V for { Ẋ = f ( (X, Z ) + ξr (t) ) f (ξ R (t)) Ż = v f (t, X, Z ) such that V and V have positive definite quadratic lower bounds near 0, and V and v f are T -periodic. Backstepping.. (5)

29 Two Other Key Assumptions We know v f and a global strict LF V for { Ẋ = f ( (X, Z ) + ξr (t) ) f (ξ R (t)) Ż = v f (t, X, Z ) such that V and V have positive definite quadratic lower bounds near 0, and V and v f are T -periodic. Backstepping.. See Sontag text, Chap. 5. (5)

30 Two Other Key Assumptions We know v f and a global strict LF V for { Ẋ = f ( (X, Z ) + ξr (t) ) f (ξ R (t)) Ż = v f (t, X, Z ) such that V and V have positive definite quadratic lower bounds near 0, and V and v f are T -periodic. Backstepping.. See Sontag text, Chap. 5. There are known positive constants θ M, ψ and ψ such that for each i {1, 2,..., s}. (5) ψ < ψ i < ψ and θ i < θ M (6)

31 Two Other Key Assumptions We know v f and a global strict LF V for { Ẋ = f ( (X, Z ) + ξr (t) ) f (ξ R (t)) Ż = v f (t, X, Z ) such that V and V have positive definite quadratic lower bounds near 0, and V and v f are T -periodic. Backstepping.. See Sontag text, Chap. 5. There are known positive constants θ M, ψ and ψ such that (5) ψ < ψ i < ψ and θ i < θ M (6) for each i {1, 2,..., s}. Known directions for the ψ i s.

32 Dynamic Feedback

33 Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s.

34 Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (7)

35 Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (7) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s

36 Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (7) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) )

37 Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (7) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) ) and i = V z i (t, ξ)u i (t, ξ, ˆθ, ˆψ). (8)

38 Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (7) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) ) and i = V z i (t, ξ)u i (t, ξ, ˆθ, ˆψ). (8) u i (t, ξ, ˆθ, ˆψ) = v f,i (t, ξ) g i (ξ) k i (ξ) ˆθ i +ż R,i (t) ˆψ i (9)

39 Dynamic Feedback The estimator evolves on { s i=1 ( θ M, θ M ) p i } (ψ, ψ) s. ˆθ i,j = (ˆθ i,j 2 θm 2 )ϖ i,j, 1 i s, 1 j p i ( ) ( ˆψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s (7) Here ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) for i = 1, 2,..., s, ϖ i,j = V z i (t, ξ)k i,j ( ξ + ξ R (t) ) and i = V z i (t, ξ)u i (t, ξ, ˆθ, ˆψ). (8) u i (t, ξ, ˆθ, ˆψ) = v f,i (t, ξ) g i (ξ) k i (ξ) ˆθ i +ż R,i (t) ˆψ i (9) The estimator and feedback can only depend on things we know.

40 Augmented Error Dynamics to be Made UGAS

41 Augmented Error Dynamics to be Made UGAS x = f ( ξ + ξ R (t)) f (ξ R (t)) z i = v f,i (t, ξ) + k i ( ξ + ξ R (t)) θ i + ψ i u i (t, ξ, ˆθ, ˆψ), 1 i s θ i,j = (ˆθ2 i,j θm) 2 ϖ i,j, 1 i s, 1 j p i ( ) ( ψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s. (10)

42 Augmented Error Dynamics to be Made UGAS x = f ( ξ + ξ R (t)) f (ξ R (t)) z i = v f,i (t, ξ) + k i ( ξ + ξ R (t)) θ i + ψ i u i (t, ξ, ˆθ, ˆψ), 1 i s θ i,j = (ˆθ2 i,j θm) 2 ϖ i,j, 1 i s, 1 j p i ( ) ( ψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s. Tracking error: ξ = ( x, z) = ξ ξ R = (x x R, z z R ) Parameter estimation errors: θ i = θ i ˆθ i and ψ i = ψ i ˆψ i Estimators: ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) and ˆψ = ( ˆψ 1,..., ˆψ s ) (10)

43 Augmented Error Dynamics to be Made UGAS x = f ( ξ + ξ R (t)) f (ξ R (t)) z i = v f,i (t, ξ) + k i ( ξ + ξ R (t)) θ i + ψ i u i (t, ξ, ˆθ, ˆψ), 1 i s θ i,j = (ˆθ2 i,j θm) 2 ϖ i,j, 1 i s, 1 j p i ( ) ( ψ i = ˆψ i ψ ˆψ i ψ) i, 1 i s. Tracking error: ξ = ( x, z) = ξ ξ R = (x x R, z z R ) Parameter estimation errors: θ i = θ i ˆθ i and ψ i = ψ i ˆψ i Estimators: ˆθ i = (ˆθ i,1,..., ˆθ i,pi ) and ˆψ = ( ˆψ 1,..., ˆψ s ) ( s { X = R r+s pi }) i=1 j=1 (θ i,j θ M, θ i,j + θ M ) ( s i=1 ( ψi ψ, ψ i ψ )) R r+s+p 1+...p s+s. (10)

44 Stabilization Analysis

45 Stabilization Analysis We build a global strict LF for the augmented error Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) X dynamics.

46 Stabilization Analysis We build a global strict LF for the augmented error Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) X dynamics. We start with this nonstrict barrier type LF on X : V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm.

47 Stabilization Analysis We build a global strict LF for the augmented error Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) X dynamics. We start with this nonstrict barrier type LF on X : V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm. On X, V 1 W ( ξ) for some positive definite function W.

48 Stabilization Analysis We build a global strict LF for the augmented error Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) X dynamics. We start with this nonstrict barrier type LF on X : V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm. On X, V 1 W ( ξ) for some positive definite function W. This is insufficient for robustness analysis because V 1 could be zero outside 0.

49 Stabilization Analysis We build a global strict LF for the augmented error Y = ( ξ, θ, ψ) = (ξ ξ R, θ ˆθ, ψ ˆψ) X dynamics. We start with this nonstrict barrier type LF on X : V 1 (t, ξ, θ, ψ) = V (t, ξ) + + s i=1 s p i i=1 j=1 ψi 0 θi,j 0 m θ 2 M (m θ i,j) 2 dm m (ψ i m ψ)(ψ ψ i + m) dm. On X, V 1 W ( ξ) for some positive definite function W. This is insufficient for robustness analysis because V 1 could be zero outside 0. Therefore, we transform V 1.

50 Transformation from Our Paper

51 Transformation from Our Paper Theorem: We can construct K K C 1 such that V (t, ξ, θ, ψ) =. K ( V 1 (t, ξ, θ, ψ) ) s + Υ i (t, ξ, θ, ψ), (11) i=1 where Υ i (t, ξ, θ, ψ) = z i λ i (t)α i ( θ i, ψ i ) + 1 T ψ α i ( θ i, ψ i )Ω i (t)α i ( θ i, ψ i ), (12) λ i (t) = ( k i (ξ R (t)), ż R,i (t) g i (ξ R (t)) ), (13) [ ] α i ( θ i, ψ θi ψ i ) = i θ i ψ i, and ψ i Ω i (t) = (14) t t t T m λ i (s)λ i (s)ds dm, is a global strict LF for the Y dynamics on X.

52 Application: BLDC Motor (Dawson-Hu-Burg)

53 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit.

54 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm.

55 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2

56 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ).

57 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ).

58 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ). y 1, y 2 = load position and velocity.

59 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ). y 1, y 2 = load position and velocity. ζ i = winding currents.

60 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ). y 1, y 2 = load position and velocity. ζ i = winding currents. B = viscous friction coefficient.

61 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ). y 1, y 2 = load position and velocity. ζ i = winding currents. B = viscous friction coefficient. M = mechanical inertia.

62 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ). y 1, y 2 = load position and velocity. ζ i = winding currents. B = viscous friction coefficient. M = mechanical inertia. N = related to the load mass and gravitational constant.

63 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ). y 1, y 2 = load position and velocity. ζ i = winding currents. B = viscous friction coefficient. M = mechanical inertia. N = related to the load mass and gravitational constant. K τ, K b = torque transmission coefficients.

64 Application: BLDC Motor (Dawson-Hu-Burg) Linear magnetic circuit. Drives single-link, direct-drive robot arm. ẏ 1 = y 2 ẏ 2 = B M y 2 N M sin(y 1) + K τ [K b ζ 1 + 1]ζ 2 (15) ζ i = H i (y, ζ)β i + γ i u i, i = 1, 2 H 1 (y, ζ) = ( ζ 1, y 2 ζ 2 ). H 2 (y, ζ) = ( ζ 2, y 2 ζ 1, y 2 ). y 1, y 2 = load position and velocity. ζ i = winding currents. B = viscous friction coefficient. M = mechanical inertia. N = related to the load mass and gravitational constant. K τ, K b = torque transmission coefficients. The unknown vectors β 1 R 2 and β 2 R 3 and unknown scalars γ 1 and γ 2 are the motor electric parameters.

65 Conclusions

66 Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering.

67 Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust.

68 Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust. Our strict Lyapunov functions gave robustness to additive uncertainties on the parameters using the ISS paradigm.

69 Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust. Our strict Lyapunov functions gave robustness to additive uncertainties on the parameters using the ISS paradigm. We covered systems with unknown control gains including brushless DC motors turning mechanical loads.

70 Conclusions Adaptive tracking and estimation is a central problem with applications in many branches of engineering. Standard adaptive control treatments based on nonstrict Lyapunov functions only give tracking and are not robust. Our strict Lyapunov functions gave robustness to additive uncertainties on the parameters using the ISS paradigm. We covered systems with unknown control gains including brushless DC motors turning mechanical loads. It would be useful to extend to cover models that are not affine in Γ, feedback delays, and output feedbacks.